Alternatively, the site could limit the number of introductions that users could send per month. For example, if there were 10 men for every 1 woman, but men could only contact one woman per month (but women could contact a new man every day), women might be willing to give those introductions more consideration.

In fact, rather than hard-coding those rate limits (or generating them algorithmically from the gender distribution of accounts) it might be viable to let each user decide on the limit themselves. It may not be a good idea to show users the specific rate limit that another user has chosen, though, as that number might give all sorts of weird signalling about how "desirable" they consider themselves to be.